Since they signed their first e-mail Barbara, then it appears that they at least sometimes professionally go by Barbara. I'd use the full name unless I got to know them more, or ended up working with them much more often via e-mail. (Or if I was explicitly told, "Please call me Barb".)

ETA: I don't think there's an "official" right or wrong in this. The only wrong would be if someone called her Barb and she hadn't even signed it Barb. If, after she signed it Barb, someone felt comfortable using that I wouldn't think they made the wrong choice. I, however, would feel more comfortable sticking with the full name for a bit longer.

I think if she used Barb then clearly Barb is fine, although Barbara would still be as well.

I always seem to get people responding with misspellings of my nickname.ie, I sign off as Niki and they respond to Nikki. Repeatedly. Drives me nuts! It is written RIGHT THERE in front of you! I don't care how you feel my name should be spelled.

I think if she used Barb then clearly Barb is fine, although Barbara would still be as well.

I always seem to get people responding with misspellings of my nickname.ie, I sign off as Niki and they respond to Nikki. Repeatedly. Drives me nuts! It is written RIGHT THERE in front of you! I don't care how you feel my name should be spelled.

I know how you feel; I feel the same way. It particularly annoys me when I've spelled my name correctly for them already, and they continue to misspell it, especially if they want something from me, i.e. to try to sell me something, or they REALLY should know by now how to spell my name, such as my doctor or other such professional whose services I depend on...it does not give me a very secure feeling!

I think if she used Barb then clearly Barb is fine, although Barbara would still be as well.

I always seem to get people responding with misspellings of my nickname.ie, I sign off as Niki and they respond to Nikki. Repeatedly. Drives me nuts! It is written RIGHT THERE in front of you! I don't care how you feel my name should be spelled.

I know how you feel; I feel the same way. It particularly annoys me when I've spelled my name correctly for them already, and they continue to misspell it, especially if they want something from me, i.e. to try to sell me something, or they REALLY should know by now how to spell my name, such as my doctor or other such professional whose services I depend on...it does not give me a very secure feeling!

Ditto! For my name, I'll continue to use Niki & Nikki as an example. Let's say Niki is the more popular version of my nickname, therefore everyone wants to spell it that way even though it's spelled out as Nikki for them! Also, I have been noticing the spelling of Nikki becoming more popular. In fact there's two other Nikkis at work and everyone seems to know that they both spell their names as Nikki. I mostly get referred to as Niki but sometimes Nikki. I don't want to make a big deal out of it so I just continue spelling my nickname as it should be spelled in my e-mails.

Also, when I was in college there was this one guy who thought it was ridiculous that I used the spelling of Nikki instead of Niki. He used to tease me. It was funny at first but the fact that he made it an ongoing thing became really irritating. He started calling me "Nikki with 2 Ks" and even addressed me as such in a birthday card he got for me. It was a small college and other than that he was a funny and nice guy so I just rolled my eyes every time he taunted me about my nickname.

If she signs off on an email to you with her nickname, then you are clear to use that nickname going forward. If she didn't want you to use it, she wouldn't sign off with it.

This rule applies unless you receive a clearly personal, non work related email from her with the nickname. THEN I would take it as a signal that off-work discussions can use her nickname, but professional discussions should use her 'official' name.